At least half of
all stars in the sky are actually multiple star systems, with two or more
stars in orbit about their common center of mass. In this lecture
we will be concerned with systems that are close enough to significantly
interact over their lifetimes. Generally, such stars will exert a
tidal influence on each other, causing internal motions in the stars that
dissipates orbital and rotational energy until the two are tidally
locked, always facing each other. In this situation, the
system rotates rigidly in space, and although the stars may be distorted,
they are no longer losing energy due to the tidal interation.

To think about the environment
near these stars, consider a "test" mass, also rotating rigidly with the
system. Placing the center of mass of the system at the origin, the
small mass will feel the gravitation force from both stars, and also the
fictitious "centrifugal force" due to its own rotation with the system

Fc
= mw2r

Rather than work with forces,
we will consider things from the point of view of potential energy (recall
that F= -dU/dr),
so the fictitious centrifugal force on the mass is

Uc = -

mw2rdr
=

-1/2 mw2r
2.

Thus, the total potential energy
in this rotating frame, shown in the figure below, is the gravitational
potential energy plus the centrifugal potential energy, or

U = - GM1m/s1-
GM2m/s2-
1/2
mw2r
2.

Corotating coordinates for
a binary star system

For convenience, we can write
this as an effective potential energy per unit
mass, F,
[erg / g]

F = -
G(M1/s1-
M2/s2)-
1/2w2r
2.

Note that the distances s1
and s2
can be written in terms of r1
and r2
through the law of cosines, and the angular frequency of the orbit comes
from Kepler's third law for the orbital period

w2
= (2p/P)2
= G(M1+M2)/a3.
= G(M1+M2)/(r1+r2)3.

Finally, from the definition
of the center of mass, we have

M1r1
= M2r2

Combining all of these expressions
allows us to evaluate the effective gravitational potential at every point
in the orbital plane of the binary star system. Figures 17.2 and
17.3 show the results, from which we can see the shape of the equipotential
surfaces (Fig. 17.3) and understand the locations and importance
of the Lagrangianpoints, especially
the inner Lagrangian point, L1.
Further
discussion, with pictures.

As one star fills its Roche
lobe, mass can be transferred from the star to the other star
through the L1
point. The matter that is transferred goes into an accretion disk,
in which the matter loses energy (through viscous forces not well understood)
and spirals down onto the surface of the star. One scenario for the
evolution of the close binary system is shown below:

This class of outburst
in brightness is thought to be due to quasi-periodic brightenings in the
accretion disk around a white dwarf. During quiescent periods, the
mass loss rate from the inflated secondary star is about 10-10
solar masses per year. Episodically, the mass transfer from the secondary
goes up by a factor of 100, to 10-8
solar masses per year. SS
Cygni is a prototype of this kind of system.

Classical Novae

In this type of
outburst, it is the matter actually falling onto the primary white dwarf
that powers the explosion. It requires a mass transfer rate of about
10-9-10-8
solar masses per year, which over 104-105
y accumulates in a layer on the surface of the white dwarf in the form
of hydrogen rich electron-degenerate matter. This layer eventually
reaches a temperature of a few million degrees, and with the help of CNO
from the white dwarf, initiates run-away nucear burning that causes much
of this outer atmosphere to expand
into a shell. The nova outbursts come in two types, fast and
slow. Both have an equally rapid rise, but fast novae are within
2 magnitudes of maximum brightness for only a week or two, while slow novae
may take nearly 100 days for a similar decline in brightness. About
2-3 are seen in our galaxy every year, but about 30 are seen in the Andromeda
Galaxy each year, so most of those occuring in our galaxy are probably
obscured by intervening dust. The speed of ejection can be measured
from doppler shifts during the outburst, and later the expanding nebula
can be used to determine the distance to the nova (recall that transverse
velocity is related to proper motion and distance by vt
= 4.74m" dpc).

Supernovae

There are two basic
types of supernovae, classified according to observational characteristics
of their spectra near maximum brightness. Those that show no strong
hydrogen lines are Type I, those that do show strong hydrogen lines are
Type II. One type is the result of core collapse in a giant star,
while the other involves a white dwarf in a close binary system.
(Can you identify which is which?)

A recent, famous Type II
supernova was Supernova
1987A for which we can look at detailed timing
of events. Type II supernovae are responsible for all of the
elements heavier than iron in the universe, through nucleosynthesis.

There are two models for
Type I supernovae, although our text prefers the first scenario:

A carbon-core white dwarf accretes
matter from a companion until it reaches around 1.3 solar masses.

At that mass, the core suddenly
reaches sufficient pressure to begin fusing C into heaver elements.

Just as in the Helium flash,
the degenerate C cannot expand as temperature goes up, so there is a complete
conflagration
of the core until the degeneracy is explosively broken and the star releases
such energy that it is completely destroyed.

The second scenario starts the
same as the first, but the mass eventually exceeds the Chandrasekhar Limit
of 1.4 solar masses.

Supernovae as standard candles

Type I supernovae
are subdivided into three classes, Type Ia,b, and c, depending on the presence
of certain spectral lines (Si, He). Type Ia supernovae are the "cleanest"
and have remarkably constant peak brightness at a whopping -20 absolute
magnitude. Because of this, they can be used, when seen in distant
galaxies, to determine the distances to these galaxies. This is a
critical observation, since it allow us to calibrate one of our distance
scale measurements, the redshift of galaxies, which we will show later
implies that the universe is expanding.